[tex]\cos(2u) = 1-2\sin^2 u \\ \\ 2u\to u \Rightarrow u \to \dfrac{u}{2}\\ \\ \cos(u) = 1-2\sin^2\Big(\dfrac{u}{2}\Big) \Rightarrow 1-\cos(u) =2\sin^2\Big(\dfrac{u}{2}\Big) \\ \\ \lim\limits_{x\to 0}\dfrac{\sqrt{1-\cos (x^2)}}{1-\cos x} =a \\ \\ \lim\limits_{x\to 0}\dfrac{\sqrt{2\sin^2\Big(\dfrac{x^2}{2}\Big)}}{2\sin^2\Big(\dfrac{x}{2}\Big)} =a[/tex]
[tex]\lim\limits_{x\to 0}\dfrac{\sqrt 2}{2}\cdot \dfrac{\sin\Big(\dfrac{x^2}{2}\Big)}{\sin^2\Big(\dfrac{x}{2}\Big)} =a \\ \\ \dfrac{\sqrt{2}}{2}\cdot \lim\limits_{x\to 0}\dfrac{\sin\Big(\dfrac{x^2}{2}\Big)}{\sin^2\Big(\dfrac{x}{2}\Big)}\cdot \dfrac{\Big(\dfrac{x}{2}\Big)^2}{\dfrac{x^2}{2}}\cdot 2 =a \\ \\ \dfrac{\sqrt 2}{2}\cdot 2 = a \Rightarrow \boxed{a = \sqrt 2}[/tex]
